
and higher order terms vanish since they are of order \(\frac{\langle k^4 \rangle }{N^3}=\frac{O(N^2)}{N^3}\xrightarrow{}0\) . We note that Eq. REF  is consistent with a previous analysis of the same system [1]}, but we repeat calculations to highlight how number fluctuations affect the result, and because we use the same analysis for active particles.
In Fig. REF  (blue curve), we plot the transition of \(\langle W_{\mathrm {ideal}} \rangle \)  from \(k_B T \ln 2\)  for \(N=1\)  to \(k_B T / 2\)  for \(N\gg 1\) . The maximal efficiency of the system is \(\eta = \langle W_{\mathrm {ideal}}\rangle /k_B T I = 1\) , achieved for one particle. Here \(I=\ln 2\)  is the Shannon information gained from the measurement.
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